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The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside the space of level 9 forms is evident, but that it appears in a different position in this space, in a different guise, as $P2/P1$, seems to me a mystery. Here's a related mystery in level 25. I'm quite certain of my conjecture, and even how to prove it, but would be grateful for any insight as to why it's true.

NOTATION

  1. As in the level 9 case, $F$ in $Z/2[[x]]$ is $x+x^9+x^{25}+x^{49}+\ldots$, but now $G=F(x^5)$, $D=x+x^9+x^{49}+x^{81}+\ldots$ and $E=x+x^4+x^9+x^{16}+x^{36}+\ldots$ where the exponents in $D$ and $E$ are the squares prime to 10 and 5 respectively. $S$ and $T$ are $(E^{16})*G$ and $(E^8)*G$.

  2. $Z/2[G^2]$ is a subring of $Z/2[[x]$]; view the latter as a module over the former. Define submodules $P1$ and $P2$ of $Z/2[[x]]$ of ranks 4 and 6 as follows: A basis of $P1$ consists of $D1=D$, $D3=(D^8)/G$, $D7=(D^2)*G$ and $D9=(D^4)*G$. P2 is generated by $P1$, $S$, and $T$. It can be shown, using modular forms of levels 5 and 25, that $P1$ and $P2$ are "Hecke-stable", i.e. that they're stabilized by the $T_p$ with $p$ not 2 or 5. Let $H=F(x^{25})$.

CONJECTURE

Let $P3$ be the $Z/2[G^2]$-module of rank 10 generated by $ P2, Y1, Y3, Y7$, and $Y9$, where $Y9=(D^2)*(G)*(G+H)^2$, $Y7=T_7(Y9)$, $Y3=T_3(Y9)$, and $Y1=T_3(Y3)$. Then:

(a) $P3$ is Hecke-stable

(b) The $Z/2[G^2]$-linear isomorphism $P1\to P3/P2$ that takes $Di$ to $Yi$, $i=1,3,7,9$ preserves the Hecke action.

NOTE

There is as far as I can see no a priori reason for believing (a) or (b). But the computer insists that they're true and I'm sure that an ad hoc proof, along the lines I sketched in my earlier question, can be cobbled out.

QUESTION

What lies behind the above isomorphism (and the similar isomorphism in level 9)?

EDIT--A FURTHER MIRACLE

Let P be the subspace of Z/2[[x]] consisting of mod 2 modular forms f for Gamma_0 (25) such that all exponents appearing in f are prime to 10. P is contained in Z/2(F,G,H) where H is G(x^5). If f lies in P one writes f as f(plus)+f(minus) where all exponents k appearing in f(plus) (resp. f(minus)) are squares (resp. non-squares) mod 5. f(plus) and f(minus) lie in P.

Let P# consist of those f in P such that the traces of f(plus) and f(minus) from Z/2(F,G,H) to Z/2(F,G), i.e. from level 25 to level 5, are both 0. P# is Hecke-stable and stable under multiplication by G^2. As a Z/2[G^2] module it has rank 18.

Now there are two mod 2 eigensystems of level 25, the first attached to delta, and the second to a weight 4 level 25 newform. Accordingly P# is a direct sum of two generalized eigenspaces.I see no reason why either of these spaces should be stable under multiplication by G^2. But now the miracle occurs.

The power series D1, D3, D7, D9, S, T, Y1, Y3, Y7, and Y9 that I introduced in my question all lie in P#. So the rank 10 Z/2[G^2] module P3 is contained in P#, and using its Hecke structure one finds that it is a subspace of the first generalized eigenspace.

I'm now convinced that I can write down 8 elements of the second generalized eigenspace that generate a complement to P3 in P#, and show that this complement is Hecke-stable and contained in the second generalized subspace.

So I can decompose P# explicitly, and show that P3 is in fact the entire first generalized eigenspace.This "explains" why P3 is Hecke-stable, only to raise further questions. Why does P3 have its peculiar Hecke-module structure? Why are the generalized eigenspaces in this situation stable under multiplication by G^2? And does any of this curious level 9 and level 25 theory generalize to other levels and characteristics?

EDIT___THE LEVEL 27 SITUATION

The analogous situation in level 27 is illuminating, though it requires a bit of preliminary notation. As usual, F in Z/2[[x]] is x+x^9+x^25+ ... , the exponents being the odd squares. Now G,H and J will be F(x^3), F(x^9) and F(x^27). F,G,H and J are the mod 2 reductions of delta(z), delta(3z), delta(9z) and delta(27z). It turns out that one wants to view the space M(27,odd) of odd mod 2 modular forms of level Gamma_0 (27) as a module over Z/2[(G+H)^2]; the point being that the Fricke involution W_27 fixes G+H. As such a module, M(27,odd) is free of rank 18. Though I haven't written out full details I believe I can show:

M(odd,27) is a direct sum of 3 Hecke stable submodules of ranks 4, 4, and 10. These are the generalized eigenspaces corresponding to the three mod 2 eigensystems of level Gamma_0 (27). (Alex Ghitza has shown that there are only three mod 2 eigensystems of this level).

One first wants to work with M(27,odd,small), the rank 9 Hecke stable submodule of M(27,odd) consisting of elements fixed by W_27. This has a decomposition into Hecke stable submodules of ranks 2,2, and 5, and it shouldn't be hard to get from this result to the result for the full space M(27,odd).

Explicitly one may construct mod 2 modular forms t and B of level 27 with the following properties:

a) t is the reduction of the level 27 weight 2 cusp form, and t^3 = G+H.

b) t*(B^4) is the reduction of a certain level 108 weight 4 newform, and 1+(B^3) = t*B.

c) A Z/2[t^6]- basis of M(27,odd,small) consists of t, t^5, t* (B^4), (t^5)*(B^2),

t^3, (t^3)* (B^2), (t^3)* (B^4), t* (B^2), and (t^5)* (B^4).

d) The first 2 elements in this basis generate the generalized eigenspace corresponding to the eigensystem attached to t. The third and fourth elements generate the generalized eigenspace corresponding to the eigensystem attached to t*(B^4). The last 5 elements generate the eigenspace corresponding to the level 1 eigensystem.

Finally I give precise descriptions of t and B as power series and an outline of the argument. t is F + J + (F*J)^ (1/4). B is 1+ E(x)+ E(x^3), where E is the level 9 modular form x+x^4+x^16+x^25 ... , the exponents being the squares prime to 3. Since W_27 interchanges F and J and also interchanges E(x) and 1+E(x^3),it fixes t and B.

Using recursions attached to T_5 and T_7 one shows that the submodule generated by t and t^5 is stabilized by T_5 and T_7 + I, and that these operators are degree decreasing. Results of a similar sort hold for the second submodule (and the operators T_5 + I and T_7 + I) and for the third submodule (and the operators T_5 and T_7). This gives the stability of the three submodules under the T_p for all p>3, and since there are only 3 generalized mod 2 eigensystems of level 27, our spaces are just the generalized eigenspaces attached to these eigensystems.

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Like the corresponding level 9 question, this is best understood using the Fricke involution. Let M(odd) be the space of odd mod 2 modular forms of level Gamma_0 (25), and J (I call it P above) be the kernel of U_5 acting on M(odd); it is a Z/2[G^2] module of rank 24 stabilized by the T_p. (Throughout p is an odd prime other than 5.) In part A of this answer I state 2 theorems that give a Hecke stable filtration J > J2 > J1 > 0 of J by Z/2[G^2]-modules, with J1 and J2 of ranks 10 and 20, and J2/J1 Hecke-isomorphic to J1. I then indicate how to get results of the desired sort, using nothing but the Hecke operator T_3.

The proofs of Theorems 1 and 2 are based on the Fricke involution and are sketched in part B. It's clear that the method will give some sort of analog whenever the level is an odd square.

PART A

F, G, H, D, and E are as in the question. D1, D3, D7 and D9 are D, D^8/G, (D^2)G and (D^4)G. S is D^5 + G,T is S^2/G. B1, B3, B7 and B9 are D^6/G, D^3, D^12/G and D^9. These are all elements of J. f-->f* is the involuton of the space of mod 2 modular forms of level Gamma_0 (25) induced by the Fricke involution of that level; it interchanges F and H, fixes G and D=F+H, is Z/2[G]-linear and commutes with the T_p.

Let s: J--->Z/2[[x]] be the map f--->f+f*.

Theorem 1

The kernel J1 of s (which evidently is a Hecke-stable Z/2[G^2]-submodule of J) has rank 10 and is generated by D1, D3, D7, D9, S, T, B1, B3, B7, B9.

Theorem 2

Let (J1,G) be the Z/2[G^2]-module generated by J1 and G. Let J2 consist of all f in J with s(f) in J1; it evidently is a Hecke-stable Z/2[G^2]-module of J containing J1. Then s maps J2 onto (J1,G)/(G) (which identifies with J1) and so gives a Hecke isomorphism between J2/J1 and J1.

Now let J1a be the Z/2[G^2] submodule of J1 generated by D1, D3, D7, D9, S, T. Level 5 theory shows that it is Hecke stable and that the T_p act locally nilpotently on it. On the other hand, B1, B3, B7 and B9 are killed by a power of I+T3; for example T_3 takes B3 to B1 and B1 to B3.This suggests looking at the Z/2[G^2]-submodule J1b of J1 generated by B1, B3, B7, B9. Let B_1, B_3, B_7, B_9 be B1, B3, B7 and B9; let B_k+10 be (G^2)B_k, and b_k be T_3(B_k). There is a recursion connecting b_k+40, b_k+10 and b_k. Using this and a calculation of initial values one can show that T_3 stabilizes J1b and that I+T_3 acts locally nilpotently on it. It follows that T_3(I+T_3) acts locally nilpotently on J1 with generalized eigenspaces J1a and J1b. So J1b like J1a is stabilized by the T_p.

Theorem 2 now shows that T_3(1+T_3) acts locally nilpotently on J2 as well as on J1. So we get a decomposition of J2 into generalized eigenspaces J2a and J2b for T_3. These are Hecke-stable. An argument like that carried out for J1b, using T_3 should show that they are in fact Z/2[G^2]-modules, and will give identifications of J2a/J1a with J1a, and of J2b/J1b with J1b as Hecke-modules.

PART B

I'll make use of the irreducible equation for D over Z/2(G):

(*) D^15 + (G^4)(D^3) + G^3 = 0

This shows that (D^5)/G is a root of z^4+z=D^8. Now evidently E^4+E=D. So E^8 and ((D^5)/G)+1 are each roots of this equation. Comparing constant terms we see that ((D^5/G)+1 =E^8. Then S and T are (E^8)*G and (E^16)*G. Since E is the reduction of a weight 2 Eisenstein series of level Gamma_0 (25), it is mod 2 modular of that level and the same holds for S and T. Examining the exponents appearing in (E^8)*G and (E^16)*G we find that S and T lie in J.

Since f-->f* fixes D and G, S and T are in J1. We now derive results (1) (2) (3) (4) that will be used to prove Theorems 1 and 2 stated in part A.

(1) Besides S and T, D1, D3, D7, D9, B1, B3, B7, B9 are in J1. These 10 elements generate a rank 10 Z/2[G^2] module.

Proof: (D^5)/G and D are mod 2 modular forms of level Gamma_0 (25), so the same holds for (D^k)/G, k in {6,8,12}. It is easily verified that the exponents appearing in the Bj and Dj are prime to 10. So the Bj and Dj lie in J; since f-->f* fixes D and G they are in J1. (*) above shows that 1, D, D^2, ... , D^14 are linearly independent over Z/2(G), giving the final assertion.

(2) The 10 elements of (1), with S and T replaced by D^5=S+G and (D^10)/G=T+G, all lie in s(J).

Proof: s(D^2*G*H^2)=D^4*G=D9. Now define D_k for k prime to 10, by taking D_1, D_3, D_7, D_9, to be D1, D3, D7, D9, and D_k+10 to be G^2*D_k. Calculating the effect of T_3 on the D_k isn't hard--we find that T3 takes D9 to D3, D3 to D1, and D_21 = G^4*D1 to D7. But T_3 and multiplication by G^4 stabilize s(J). Also s takes D*H^4 and D^2*H^8/G to D^5=S+G and D^10/G=T+G. Furthermore s takes D*H^2,D^8*H^4/G and D*H^8 to D^3=B3, D^12/G=B7, and D^9=B9.

And G*(T_3(D^3)) and D^6 are reductions of weight 24 holomorphic modular forms of level Gamma_0 (25). One verifies that the power series G*T_3(D^3) and D^6 agree through exponent (30)(24)/12=60, the Sturm bound, and conclude that B1=T_3(B3).

(3) D^7, D^11, D^13, and D^14/G lie in s(J). Together with the 10 elements of (2) they generate a rank 14 Z/2[G^2]-submodule of s(J).

Proof: s takes D^3*H^4, D^9*H^2, D^9*H^4, and D^6*H^8/G to D^7,D^11,D^13 and D^14/G. The final assertion comes from (*).

Now J has Z/2[G^2]-rank 24. Combining (1) and (3) we find that J1 and s(J) have ranks 10 and 14. Now we've seen that D^5/G is a root of z^4+z=D^8. Since D lies in the integral closure, R, of Z/2[G] in Z/2(D,G), so does D^5/G. It follows that the following u_0,u_1, ... ,u_14 all lie in R: 1, D, D^2, D^3, D^4, D^5/G, D^6/G, D^7/G, D^8/G, D^9/G, D^10/G^2, D^11/G^2, D^12/G^2, D^13/G^2. D^14/G^2.

(4) The above u_i are a Z/2[G]-basis of R.

This I suppose can be verified by computer. I have a fairly short proof by hand that I can put up if anyone wants to see it.

Theorem 1: The 10 elements of (1) are a Z/2[G^2] basis of J1.

Proof: Suppose u is in J1. Since J1 has Z/2[G^2]-rank 10, u is a Z/2(G) linear combination of these 10 elements and lies in Z/2(D,G). As the ring of mod 2 modular forms of level Gamma_0 (N) is integrally closed, u lies in R. It follows from (4) and the fact that u is odd that it is a Z/2[G^2] linear combination of G, D, D^2*G, D^4*G, D^8/G, S, T, D^6/G, D^3, D^12/G, D^9, D^7, D^11/G^2, D^13/G^2, and D^14/G. Since u is in J1, the coefficients of the last 4 elements are 0. Finally all the remaining elements, apart from G, lie in J1. So the coefficient of G is 0 as well, giving the result.

Theorem 2: Let J2 consist of those f in J with s(f) in (J,G). Then s:J2---> (J1,G)/(G)= J1 is onto with kernel J1. So it gives rise to a Z/2[G^2]-linear bijection of J2/J1 with J1 preserving the Hecke action.

Proof: The calculations in (2) show that the Dj, Bj, S+G and T+G are in the image. So by Theorem 1, the map is onto. If the kernel were larger than J1, s(J) would have to contain a non-zero element of the Z/2[G^2]-module (G) generated by G. But (3) and the fact that 1, D, ..., D^14 are linearly independent over Z/2(G) preclude this.(For otherwise s(J) would have rank 15, and we've shown that the rank is 14).

4/13 EDIT--A PRECISE DESCRIPTION OF J2b

I haven't yet described J2b--the generalized eigenspace for T_3 acting on J2 with eigenvalue 1--in detail. To my eye the results and their proofs are attractive and deserve an exposition. But first I'll take on J1b.

(1)___ T_3 takes:____ B_1, B_3, B_7, B_9, to B_3, B_1, B_13, B_11 + B_3

__B_11, B_13, B_17, B_19, to B_9, B_7, B_19, B_17 + B_9

__B_21, B_23, B_27, B_29, to B_23 + B_7, B_21 + B_13, B_33 + B_17 + B_9, B_31

__B_31 ,B_33, B_37, B_39, to B_29 + B_21, B_27 + B_19 + B_11, B_39 + B_31 + B_23,

_ and B_37 + B_21 + B_13.

(2)___ T_3(B_k+40) = (G^8)(T_3(B_k)) + (G^2)(T_3(B_k+10))

Remarks: (1) is proved using the Sturm bound. For example B_39 comes from a characteristic zero form of weight 108. In this weight and level, the Sturm bound is 30*108/12 = 270. So to show that T_3(B_39) and B_37 + B_21 + B_13 are equal it's enough to show that their expansions coincide through x^270. This requires expanding B_39 through x^810, which is quickly done with computer. A standard technique used by Nicolas, Serre, and me is used to get (2).

From (1) and (2) we see that T_3 stabilizes the Z/2[G^2] submodule of J1 having basis {B_1,B_3,B_7,B_9}. And also that T_3 (B_k) is a sum of B_j with each j congruent to 3k mod 8. Furthermore the largest j appearing in the sum is k+2 if k is 1, 9, or 17 mod 20, k-2 if k is 3, 11, or 19 mod 20, k+6 if k is 7 mod 20, and k-6 if k is 13 mod 20. From this it follows that (T_3)^2 takes B_k to B_k + a sum of B_j with j < k. Consequently (T_3 + I)^2 takes B_k to a sum of B_j with j < k, and T_3 + I acts locally nilpotently on our module. As T_3 acts locally nilpotently on the module with basis {D_1,D_3,D_7,D_9,S,T} and these two modules span J1, they are J1b and J1a. We next turn to J2b.

(3)___ There is a w = x + ... in Z/2[[x]] with 1 + w + w^2 = 1/(1 + E). w is the reduction of a weight 4 characteristic zero form. The characteristic 2 Fricke involution of level Gamma_0 (25) fixes E but takes w to w+1.

We now define elements AB_k of J for k > 0 and prime to 10. We take AB_1, AB_3, AB_7, and AB_9 to be (w^16)B_1, (w^8)B_3, (w^16)B_7 + D_7, and (w^32)B_9 + D_9. We set AB_k+10 equal to (G^2)(AB_k). One shows that w^4 is (G+H)/D, and it follows from the definitions that the exponents of x appearing in the AB_k are prime to 10, so that the AB_k are in J. Since our Fricke involution fixes B_k, D_k, and G, and takes w to w+1, s(AB_k) is B_k; as B_k is in J1, AB_k is in J2.

(4)___As noted above T_3 takes B_39 to B_37 + B_21 + B_13. The same formula holds when B is replaced by AB provided we put in one extra term; T_3 takes AB_39 to AB_37 + AB_21 + AB_13 + B_13. Indeed there are such results for T_3 applied to each of AB_1, AB_3,... through AB_39. The "extra terms" we get in these 16 cases are 0,0,0,B_3,0,0,0,B_9,B_7,B_13,B_17,0,B_21,B_19,B_31 and B_13.

(5)___The recursion of (2) holds with B replaced by AB throughout.

Remarks___ From (4) and (5) we see that T_3 stabilizes the Z/2[G^2] submodule of J2 with basis {B_1,B_3,B_7,B_9,AB_1,AB_3,AB_7,AB_9}. And one finds as in the argument based on (1) and (2) that T_3 + I acts locally nilpotently on the quotient of this module by J1b. So the module is contained in J2b. With a bit more work (writing down 6 elements of J2 mapping by s to generators of J1a, modifying these elements if necessary by elements of J1, so they lie in J2a, and showing using recursions that the resulting rank 12 module is stabilized by T_3, and that T_3 acts locally nilpotently on it), we conclude that our rank 8 and rank 12 modules are precisely J2b and J2a.

A bit more about (4) and its proof, with a description of the space of mod 2 forms of level Gamma_0 (25) that may be of interest. This space of forms has a filtration M(0) < M(4) < M(8) < ... where M(4k) consists of reductions of characteristic zero forms of weight 4k. Then M(4k) has dimension 10k + 1. Let alpha be 1 + w + w^2. Then a Z/2 basis of M(4k) is given by the (w^i)/(alpha^k) with i in [0,10k]. Let rho be 1 + alpha^3. Here are some formulas:

__D_1 = rho/(alpha^4), D_3 = (rho^3)/(alpha^20), D_7 = (rho^7)/(alpha^20), D_9 = (rho^9)/(alpha^28), G = (rho^5)/(alpha^12)

__B_1 = rho/(alpha^12), B_3 = (rho^3)/(alpha^12), B_7 = (rho^7)/(alpha^36), B_9 = (rho^9)/(alpha^36).

___ So for example AB_39 = (G^6)(w^32)(B_9) + (G^6)(D_9). This is the sum of (w^32)(rho^39)/(alpha^108) and (rho^39)/(alpha^100). Since 32 + (6)(39) < 270, AB_39, like B_39, lies in M(108), and one only needs as in the corresponding argument calculating T_3(B_39) to expand it up to x^810 to get the sixteenth identity in (4).

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